Tacrolimus and Mycophenolate Mofetil With or Without Sirolimus in Preventing Acute Graft-Versus-Host Disease in Patients Who Are Undergoing Donor Stem Cell Transplant for Hematologic Cancer

Inclusion Criteria: * Ages \> 50 years with hematologic malignancies treatable by unrelated hematopoietic cell transplant (HCT) * Ages =\< 50 years of age with hematologic diseases treatable by allogeneic HCT who through pre-existing medical conditions or prior therapy are considered to be at high risk for regimen related toxicity associated with a conventional transplant (\> 40% risk of transplant related mortality \[TRM\]) (This criterion can include patients with a HCT-comorbidity index (CI) score of \>= 1; transplants should be approved for these inclusion criteria by both the participating institutions' patient review committees such as the Patient Care Conference (PCC) at the Fred Hutchinson Cancer Research Center (FHCRC) and by the principal investigators at the collaborating centers) * Patients =\< 50 years of age who have received previous high-dose transplantation do not require patient review committee approvals (All children \< 12 years must be discussed with the FHCRC principal investigator (PI) \[Brenda Sandmaier, MD 206 6674961\] prior to registration) * Ages =\< 50 years of age with chronic lymphocytic leukemia (CLL); these patients do not require patient review committee approvals * Ages =\< 50 years of age with hematologic diseases treatable by allogeneic HCT who refuse a conventional HCT (Transplants must be approved for these inclusion criteria by both the participating institutions' patient review committee such as PCC at the FHCRC and by the principal i…